The Annals of Probability

Entropic measure and Wasserstein diffusion

Max-K. von Renesse and Karl-Theodor Sturm

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Abstract

We construct a new random probability measure on the circle and on the unit interval which in both cases has a Gibbs structure with the relative entropy functional as Hamiltonian. It satisfies a quasi-invariance formula with respect to the action of smooth diffeomorphism of the sphere and the interval, respectively. The associated integration by parts formula is used to construct two classes of diffusion processes on probability measures (on the sphere or the unit interval) by Dirichlet form methods. The first one is closely related to Malliavin’s Brownian motion on the homeomorphism group. The second one is a probability valued stochastic perturbation of the heat flow, whose intrinsic metric is the quadratic Wasserstein distance. It may be regarded as the canonical diffusion process on the Wasserstein space.

Article information

Source
Ann. Probab., Volume 37, Number 3 (2009), 1114-1191.

Dates
First available in Project Euclid: 19 June 2009

Permanent link to this document
https://projecteuclid.org/euclid.aop/1245434030

Digital Object Identifier
doi:10.1214/08-AOP430

Mathematical Reviews number (MathSciNet)
MR2537551

Zentralblatt MATH identifier
1177.60066

Subjects
Primary: 60J60: Diffusion processes [See also 58J65] 60G57: Random measures 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 47D07: Markov semigroups and applications to diffusion processes {For Markov processes, see 60Jxx} 60H15: Stochastic partial differential equations [See also 35R60] 60J45: Probabilistic potential theory [See also 31Cxx, 31D05] 58J65: Diffusion processes and stochastic analysis on manifolds [See also 35R60, 60H10, 60J60]

Keywords
Wasserstein space optimal transport entropy Dirichlet process change of variable formula measure-valued diffusion Brownian motion on the homeomorphism group stochastic heat flow Wasserstein diffusion entropic measure

Citation

von Renesse, Max-K.; Sturm, Karl-Theodor. Entropic measure and Wasserstein diffusion. Ann. Probab. 37 (2009), no. 3, 1114--1191. doi:10.1214/08-AOP430. https://projecteuclid.org/euclid.aop/1245434030


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